A Gaussian wave packet at time
can be represented by
is the initial width of the wave packet and
is the momentum.
At a later time
, the wave packet evolves with
is a constant,
is the mass of the particle, and
is a parameter that determines the corresponding width of the wave packet (
is the actual width). The parameter
is given by
Clearly, the width increases with time
, as the wave packet spreads. For simplicity,
have been set equal to unity.
The probability amplitude is a Gaussian function centered about the point
The wave packet maintains a Gaussian shape, with changing centroid and width.
Snapshot 1: small initial width implies faster spread
Snapshot 2: initially broad wave packet spreads out more slowly
In the limiting case, of a wave packet initially equal to a Dirac delta function, it immediately transforms into an infinite plane wave.
 H. C. Verma, Quantum Physics
, 2nd ed., Bhopura, Ghaziabad, India: Surya Publications, 2009.